We will discuss the accretivity and m-accretivity properties for a family of operators on infinite, weighted graphs. In particular, we will discuss how, for a maximal-type operator, these properties are equivalent and can be studied through the injectivity of the shifted operator. For a minimal-type operator, this equivalence is no longer true as can be seen by examples. We will also discuss connections to essential self-adjointness and stochastic completeness. This is joint work with Davide Bianchi, Matthias Keller and Alberto Setti.